Let the operation $\bigstar$ be defined by $x\bigstar y=y^x-xy$. Calculate $(3\bigstar4)-(4\bigstar3)$.

Let $ABC$ be an acute-angled triangle such that $|AB|<|AC|$. Let $X$ and $Y$ be points on the minor arc ${BC}$ of the circumcircle of $ABC$ such that $|BX|=|XY|=|YC|$. Suppose that there exists a point $N$ on the segment $\overline{AY}$ such that $|AB|=|AN|=|NC|$. Prove that the line $NC$ passes through the midpoint of the segment $\overline{AX}$. \\ \\ (Ivan Novak)

Every cell of a $2017\times 2017$ grid is colored either black or white, such that every cell has at least one side in common with another cell of the same color. Let $V_1$ be the set of all black cells, $V_2$ be the set of all white cells. For set $V_i (i=1,2)$, if two cells share a common side, draw an edge with the centers of the two cells as endpoints, obtaining graphs $G_i$. If both $G_1$ and $G_2$ are connected paths (no cycles, no splits), prove that the center of the grid is one of the endpoints of $G_1$ or $G_2$.

Prove that : \[\pi (e-1)<\int_0^{\pi} e^{|\cos 4x|}dx<2(e^{\frac{\pi}{2}}-1)\]

There are $34$ friends are sitting in a circle playing the following game. Every round, four of them are chosen at random, and have a rap battle. The winner of the rap battle stays in the circle and the other three leave. This continues until one player remains. Everyone has equal rapping ability, i.e. every person has equal probability to win a round. What is the probability that Michael and James end up battling in the same round?

At the base of the quadrangular pyramid $SABCD$ lies the parallelogram $ABCD$. Prove that for any point $O$ inside the pyramid, the sum of the volumes of the tetrahedra $OSAB$ and $OSCD$ is equal to the sum of the volumes of the tetrahedra $OSBC$ and $OSDA$ .

This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.

Katie has a chocolate bar that is a $5$-by-$5$ grid of square pieces, but she only wants to eat the center piece. To get to it, she performs the following operations: i. Take a gridline on the chocolate bar, and split the bar along the line. ii. Remove the piece that doesn’t contain the center. iii. With the remaining bar, repeat steps $1$ and $2$. Determine the number of ways that Katie can perform this sequence of operations so that eventually she ends up with just the center piece.

There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$. For that unique $a$, find $a+U$. (Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked? [i]Proposed by Andrew Wu[/i]

Show that there are no integers $n$ such that $n^4 + 4^n$ is a prime greater than $5$.

Does there exist a positive integer $n$ such that all its digits (in the decimal system) are greather than 5, while all the digits of $n^2$ are less than 5?

Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

If $m > 0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m = $ $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ 2\qquad\textbf{(E)}\ \sqrt{5} $

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

Let $ABCD$ be a rectangle of length $m$ and width $n$, with $m, n$ positive integers. Consider a ray of light that starts from $A$, reflects with an angle of $45^o$ on an opposite side and continues reflecting away at the same angle. $\bullet$ For any pair $(m,n)$, show that the ray meets a vertex at some point. $\bullet$ Suppose $m$ and $n$ are coprime. Determine the number of reflections made by the ray of light before encountering a vertex for the first time.

Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.

Ted's favorite number is equal to \[1\cdot\binom{2007}1+2\cdot\binom{2007}2+3\cdot\binom{2007}3+\cdots+2007\cdot\binom{2007}{2007}.\] Find the remainder when Ted's favorite number is divided by $25$. $\begin{array}{@{\hspace{-1em}}l@{\hspace{14em}}l@{\hspace{14em}}l} \textbf{(A) }0&\textbf{(B) }1&\textbf{(C) }2\\\\ \textbf{(D) }3&\textbf{(E) }4&\textbf{(F) }5\\\\ \textbf{(G) }6&\textbf{(H) }7&\textbf{(I) }8\\\\ \textbf{(J) }9&\textbf{(K) }10&\textbf{(L) }11\\\\ \textbf{(M) }12&\textbf{(N) }13&\textbf{(O) }14\\\\ \textbf{(P) }15&\textbf{(Q) }16&\textbf{(R) }17\\\\ \textbf{(S) }18&\textbf{(T) }19&\textbf{(U) }20\\\\ \textbf{(V) }21&\textbf{(W) }22 & \textbf{(X) }23\\\\ \textbf{(Y) }24 \end{array}$

There is a regular $2021$-gon. We put a coin with heads up on every vertex of it. Every time, you can choose one vertex, and flip the coin on the vertices adjacent to it. Can you make all the coin tails up?

Let $ABCDEF$ be a convex hexagon of area $1$, whose opposite sides are parallel. The lines $AB$, $CD$ and $EF$ meet in pairs to determine the vertices of a triangle. Similarly, the lines $BC$, $DE$ and $FA$ meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least $3/2$.

Every cell of table $4 \times 4$ is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole figure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all $n$ such that after $n$ steps it is possible to get the table with every cell colored black.

Does there exist a solution $(x,y,z,u,v)$ in integers greater than $1998$ to the equation $x^{2}+y^{2}+z^{2}+u^{2}+v^{2}=xyzuv-65$?

Let $n$ a positive integer, $n > 1$. The number $n$ is wonderful if the number is divisible by sum of the your prime factors. For example; $90$ is wondeful, because $90 = 2 \times 3^2\times 5$ and $2 + 3 + 5 = 10, 10$ divides $90$. Show that, exist a number "wonderful" with at least $10^{2002}$ distinct prime numbers.

In a school with 5 different grades there are 250 girls and 250 boys. Each grade has the same number of students. for a competition of knowledge wants to form teams of a female and a male who are of the same grade. If in each grade there are at least $19$ females and $19$ males. Find the greatest amount of teams that can be formed.